21 years of the coffee stain

dried coffee stains, alcohol and coffee

Happy 21st birthday to the coffee stain. But there is still much for us to learn 21 years after the first paper on the coffee stain was published.

On the 23rd October, 1997, a paper was published in the journal Nature titled “Capillary flow as the cause of ring stains from dried liquid drops.” The title is in the dry style that scientific papers can be written. An alternative title could have been “How coffee stains form”*. Perhaps you would think, surely someone had known how coffee stains formed before 1997? And maybe you would go on to think: certainly 21 years later in 2018, we’d know all there was to know about the coffee stain? I hope that readers of Bean Thinking would not think “who cares about coffee stains?”, but I wonder whether it was the combination of disinterest and assuming that someone somewhere surely knew how they formed that meant it took until 1997 for anyone to ask the question: well how do they form?

Coffee is a very popular drink among scientists, though even this does not explain how popular this paper has become. A paper’s popularity can be measured in ‘number of citations’ which tells you how many times other authors have found this piece of work important enough to reference it in their own published paper. As of early November 2018, this paper has been cited nearly 3300 times. Why? Well, there seem to be at least two reasons. Firstly, it turns out that the coffee stain effect is of enormous technological relevance; it may even have been used in the manufacture of the device you are using to read this website. But secondly even now, 21 years later, we still don’t understand what is going on, there is still much to learn and some of it is some very subtle and very beautiful physics.

the droplets ready to dry

What happens when you form coffee stains using drops containing two liquids (alcohol and water) compared to just one (water)?

Very recently for example, a new paper was published in Physical Review Letters. This one was titled “Density-driven flows in evaporating binary liquid droplets“. Another exciting title, another time we’ll retitle it for the purposes of this post: “what happens when you mix alcohol with a coffee type suspension, dry it at different angles and film it drying.” Arguably this time the given title is more succinct. Why does it make a difference if you add alcohol to your coffee rather than just drink it straight (the coffee, not the alcohol)? And what happens to the resulting coffee stain?

Maybe of an evening you’ve been relaxing with a glass of wine, or something stronger, and noticed the “legs” rising up the glass. Their formation and appearance is due to the differing surface tensions between alcohol and water and the fact that alcohol evaporates more easily than water, you can read more about that effect here. The point is that because of the difference in surface tension between alcohol and water, you get a flow of liquid from areas of low surface tension (higher alcohol content) to high surface tension (high water content). And it was this that had been thought to drive coffee stain formation in droplets which were a mix of liquids, water and alcohol for example. But how do you isolate this effect from the other effect in which alcohol evaporates more quickly than water and so there are changes in density and buoyancy of the droplet?

pendulant droplets

Drying droplets upside down. The things we do for coffee science.

To answer this you could add n-butanol to the water (or coffee) rather than alcohol. Just like ethanol based alcohol (the sort you may get in gin), n-butanol has a much lower surface tension and lower density than water but unlike alcohol, it evaporates much less readily than water. So, in a water-butanol mix it will be the water that goes first, while exactly the opposite will happen for an alcohol-water mix. In a drying droplet, the liquid evaporates most quickly from the edge of the drop. Therefore, after an initial, chaotic stage (imaginatively called stage I), you will end up with a droplet that is water rich around its rim in the alcohol-water mix but n-butanol rich around the droplet edge in an n-butanol-water mix (stage II). This suggests a way that you can distinguish the flows in the drop due to surface tension effects from those due to the differences in density between water and alcohol/n-butanol.

How would you test it? One way would be to compare the droplets evaporating as if you had spilled them on the table top with droplets evaporating ‘upside-down’, as if you had tipped the table by 180° after spilling your coffee. You can then watch the flow by taking many photographs with a camera. In this way you would be able to test whether it was surface tension flow (which should be in the same direction within the drop whether the droplet is upright or suspended) with gravity driven flow which should be opposite (the drop is upside down after all).

schematic drops upright and upside down

A cartoon of the flow found in droplets of alcohol and water mix. When upright, the flow is up through the centre of the drop and down the sides. This is expected for both surface tension based flows and flows due to gravity. When upside down, the flow is still upwards through the centre of the drop but this time the drop is upside down. So this is what you’d expect if the dense water at the edge of the drop flowed downwards (gravity based) but not if the flow were dominated by surface tension effects which should be the same, relative to the drop-interface as if the drop were upright.

The authors of the study did this and found that the flow in upright drops of alcohol-water was opposite to that in n-butanol-water drops. This is what is expected both in surface tension dominated flow and in gravity dominated flow. But, when the drops were inverted, the flow within the droplet did not change absolute direction, instead it changed direction relative to the substrate (it may be helpful to see the cartoon), in both droplet types. Expected for a gravity driven flow (dense liquids move downwards), this is exactly the opposite to what would be expected with surface tension driven flow. It is sensible to conclude that the flow in drying droplets containing two liquid types is dominated by gravity, or as the authors phrased it “density-driven flows in evaporating binary liquid droplets”.

dried upside down drops

The resultant coffee stains of drops that had been suspended upside down. They seem fairly similar to the upright ones with the exception of the central dot in many of the stains. The arrow shows some coffee that spilled down the surface as the tray was flipped over.

While the authors did a lovely job of watching the flows within the droplet, what happened to the the actual coffee stain? It could prompt us to do an experiment at home. How does adding alcohol affect the appearance of a coffee stain if the drop is upright compared to if you turned the drops all upside down? What happens if the droplet is not held upside down but instead at an angle to the vertical? There are many ways you could play with this result, see what happens, have a glass of wine and see if that gives you any insight into what you see with your coffee. As ever, have fun and if you do get any interesting results, please do let me know here, on twitter or over on FB.


*The dry scientific author in me wants to point out that although catchier, the title “how coffee stains form” does not actually capture the extent of the physics nor what the paper was about (the fact that this happens more often than just in coffee) and the given title was much better. The coffee drinker in me thinks yes, but, surely we could make it all about coffee anyway…

Air raising

Small waves seen from Lindisfarne

How do clouds form? How does temperature vary with altitude, and what does coffee have to do with any of it?

You put a drop of alcohol on your hand and feel your hand get cooler as the alcohol evaporates, but what has this to do with coffee, climate and physics?

Erasmus Darwin (1731-1802) was the grandfather of Charles of “Origin of the Species” fame. As a member of the Lunar Society (so-called because the members used to meet on evenings on which there was a full moon so that they could continue their discussions into the night and still see their way home) he would conduct all sorts of scientific experiments and propose various imaginative inventions. Other members of the Lunar Society included Matthew Boulton, Josiah Wedgwood and Joseph Priestley. The society was a great example of what can happen when a group of people who are interested in how things work get together and investigate things, partly just for the sake of it.

One of the things that Darwin had noticed was that when ether* evaporates from your hand, it cools it down, just as alcohol does. Darwin considered that in order to evaporate, the ether (or alcohol or even water) needed the heat that was provided by his hand, hence his hand started to feel cooler. But then he considered the corollary, if water (ether or alcohol) were to condense, would it not give off heat? He started to form an explanation of how clouds form: As moist air rises, it cools and expands until the moisture in the air starts to condense into droplets, clouds.

hole in water alcohol

There are several cool things you can notice with evaporating alcohol. Here a hole has been created in a thin layer of coffee by evaporating some gin. You can see the video of the effect here.

As with many such ideas, we can do a ‘back of the envelope’ calculation to see if Darwin could be correct, which is where we could also bring in coffee. The arabica growing regions are in the “bean belt” between 25 °N and 30 °S. In the sub-tropical region of that belt, between about 16-24°, the arabica is best grown at an altitude between 550-1100 m (1800-3600 ft). In the more equatorial regions (< 10º), the arabica is grown between 1100-1920m (3600-6300 ft). It makes sense that in the hotter, equatorial regions, the arabica needs to be grown at higher altitude so that the air is cooler, but can we calculate how much cooler it should be and then compare to how much cooler it is?

We do this by assuming that we can define a parcel of air that we will allow to rise (in our rough calculation of what is going on)¹. We assume that the parcel stays intact as it rises but that its temperature and pressure can vary as they would for an ideal gas. Assuming that the air parcel does not encounter friction as it rises (so we have a reversible process), what we are left with is that the rate of change of temperature with height (dT/dz) is given by the ratio of the gravitational acceleration (g) to the specific heat of the air at constant pressure (Cp) or, to express it mathematically:

dT/dz = -g/Cp = Γa

Γa is known as the adiabatic lapse rate and because it only depends on the gravitational acceleration and the specific heat of the gas at constant pressure (which we know/can measure), we can calculate it exactly. For dry air, the rate of change of temperature with height for an air parcel is -9.8 Kelvin/Km.

contrail, sunset

Contrails are caused by condensing water droplets behind aeroplanes.

So, a difference in mountain height of 1000 m would lead to a temperature drop of 9.8 ºC. Does this explain why coffee grows in the hills of Mexico at around 1000 m but the mountains of Columbia at around 1900 m? Not really. If you take the mountains of Columbia as an example, the average temperature at 1000 m is about 24ºC all year, but climb to 2000 m and the temperature only drops to 17-22ºC. How can we reconcile this with our calculation?

Firstly of course we have not considered microclimate and the heating effects of the sides or plateaus of the mountains together with the local weather patterns that will form in different regions of the world. But we have also missed something slightly more fundamental in our calculation, and something that will take us back to Erasmus Darwin: the air is not dry.

Specific heat is the amount of energy that is required to increase the temperature of a substance by one degree. Dry air has a different specific heat to that of air containing water vapour and so the adiabatic lapse rate (g/Cp) will be different. Additionally however we have Erasmus Darwin’s deduction from his ether: water vapour that condenses into water droplets will release heat. Condensing water vapour out of moist air will therefore affect the adiabatic lapse rate and, because there are now droplets of water in our air parcel, there will be clouds. When we calculate the temperature variation with height for water-saturated air, it is as low as 0.5 ºC/100 m (or 5 K/Km), more in keeping with the variations that we observe in the coffee growing regions†.

We have gone from having our head in the clouds and arrived back at our observations of evaporating liquids. It is fascinating what Erasmus Darwin was able to deduce about the way the world worked from what he noticed in his every-day life. Ideas that he could then either calculate, or experiment with to test. We have very varied lives and very varied approaches to coffee brewing. What will you notice? What will you deduce? How can you test it?


*ether could refer to a number of chemicals but given that Erasmus Darwin was a medical doctor, is it possible that the ether he refers to was the ether that is used as an anaesthetic?

†Though actually we still haven’t accounted for microclimate/weather patterns and so it is still very much a ‘rough’ calculation. The calculation would be far better tested by using weather balloons etc. as indeed it has been.

¹The calculation can be found in “Introduction to Atmospheric Physics”, David Andrews, Cambridge University Press



Scratching the surface in coffee week

reflections, surface tension

The effects of surface tension can be seen in the light reflected from a coffee

UK Coffee week is once again upon us meaning that all week we can be justified in thinking about, drinking, appreciating and celebrating coffee. And of course, as soon as we start to do this, we realise we have to drink, appreciate and celebrate water which is, ultimately, what really makes most of the cup of coffee. So UK Coffee Week raises money for Project Waterfall which is a charity that brings clean water to coffee growing communities. Giving something back by enjoying something good.

In keeping with the water theme, this week The Daily Grind is all about water, including an experiment that enables you to make a hole in it. As this is also the week between Palm Sunday and Easter, perhaps we could call the post “Holey water for Holy Week”.

But moving quickly to the experiment. While drinking your coffee, you may have noticed how around the edge of the cup, the coffee appears lighter, not quite so dark, as in the interior. The coffee is being bent upwards at the edge of the cup by the surface tension of the water in the coffee. Now, what happens if you add alcohol to the coffee? If you do this in your coffee cup you may well end up with an Irish coffee which may provide even more of an excuse to celebrate your coffee drinking, but if you were to put your coffee on a plate first (I know, why? but bear with me) you will get a quite different result. You will be able to make a hole in the middle of your coffee. The reason is that the surface tension of alcohol is much weaker than that of water. Consequently, if you try to mix a very thin layer of coffee with a small amount of alcohol, something slightly unexpected happens as this video shows:

The addition of a small amount of alcohol into the middle of a thin layer of water (or coffee) causes the water to recede. As the alcohol evaporates off, you are left with a dry ‘hole’ in the coffee. Why is this? It is effectively a liquid-tug-of-war on your plate. The higher surface tension in the coffee (or water) pulls against the weaker surface tension of the alcohol which eventually means that the water breaks away, leaving the hole. As the water molecules are continually moving, eventually they start to meet again over the dry spot and close the hole.

You can’t see this in your mug of course because the mixing occurs throughout the liquid while the plate ensures that this is only a surface effect.

You will need a strong alcohol, perhaps gin or vodka but please do try this experiment, let me know how you get on and enjoy the coffee, water (and alcohol) in UK Coffee week. And if you want to donate to Project Waterfall, you could either find a participating café here or donate online here.